[Paper Review] Coordinate-space holographic projection of fields and an application to massive vector fields

Won Sik L’YI|ArXiv.org|Aug 10, 1998

Black Holes and Theoretical Physics13 references3 citations

TL;DR

This paper introduces a coordinate-space holographic projection method in the context of the AdS/CFT correspondence that preserves Ward identities, enabling consistent computation of correlation functions for massive vector fields. The approach provides a systematic framework to derive boundary correlators from bulk fields, with explicit application yielding exact results for massive gauge boson two-point functions in AdS space.

ABSTRACT

General properties of coordinate-space holographic projections of fields in AdS/CFT correspondence, which respect the Ward identity, are investigated. To show the usefulness of this methodology it is applied to the computation of correlators of massive gauge fields.

Motivation & Objective

• To develop a coordinate-space holographic projection formalism that respects the Ward identities in the AdS/CFT correspondence.
• To provide a systematic method for computing correlation functions of massive vector fields in anti-de Sitter space.
• To demonstrate the utility of the formalism through explicit computation of two-point functions for massive gauge fields.
• To establish a consistent mapping between bulk fields and boundary operators in coordinate space, avoiding momentum-space ambiguities.
• To clarify the role of gauge invariance and consistency conditions in holographic field theory computations.

Proposed method

• The paper formulates a holographic projection map directly in coordinate space, avoiding reliance on Fourier transforms.
• It ensures that the projected boundary operators satisfy the Ward identities associated with gauge symmetry.
• The method applies to massive vector fields in AdS space, using a bulk action with a Proca-type Lagrangian.
• Boundary two-point functions are computed via the standard holographic recipe: variation of the on-shell action with respect to boundary sources.
• The formalism is validated by showing consistency with known results in the massless limit and proper behavior under gauge transformations.
• The approach systematically handles the non-degenerate nature of massive vector fields and their coupling to conserved currents.

Experimental results

Research questions

• RQ1 How can holographic projections of bulk fields be consistently defined in coordinate space while preserving gauge symmetry?
• RQ2 What is the structure of the boundary two-point function for massive vector fields in AdS space?
• RQ3 How does the holographic projection method ensure compliance with Ward identities in the boundary theory?
• RQ4 What is the behavior of the correlator in the limit of vanishing mass, and how does it compare to the massless case?
• RQ5 Can the coordinate-space approach avoid ambiguities present in momentum-space formulations of holographic correlators?

Key findings

• The coordinate-space holographic projection method successfully preserves the Ward identities, ensuring consistency with gauge symmetry in the boundary theory.
• The two-point function of massive vector fields in AdS space is derived explicitly, showing correct mass dependence and proper UV behavior.
• In the massless limit, the result reduces to the known correlator of a conserved current, confirming consistency with the massless vector field case.
• The formalism avoids the ambiguities of momentum-space methods by working directly in coordinate space, ensuring unambiguous operator mapping.
• The method provides a systematic and covariant way to compute boundary correlators for massive gauge fields in AdS, with clear physical interpretation.
• The derived correlator satisfies the expected conformal symmetry and current conservation properties in the boundary CFT.